Legal claims defining the scope of protection, as filed with the USPTO.
1. A method for approximating the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α), said method comprising: (a) representing a plurality of images using an individual data array for each of said plurality of images; (b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P −1 ) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to said power (α) to produce a fractional power diagonal operator; (d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P −1 ) to produce a first partial result; (e) multiplying a data array of one of said plurality of images by said ordered similarity transformation operator (P) to produce a modified data array; (f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator; and (g) repeating operations (e) and (f) for each of said plurality of images.
2. The method according to claim 1 , wherein said data array, for each of said plurality of images, is a vector.
3. The method according to claim 1 , wherein said data array, for each of said plurality of images, is a matrix.
4. The method according to claim 3 , wherein each of said plurality of matrices represent a monochrome image.
5. The method according to claim 3 , wherein each of said plurality of matrices represent a luminance component of a represented image.
6. The method according to claim 3 , wherein each of said plurality of matrices represent a chroma component of a represented image.
7. The method according to claim 1 , wherein said data array, for each of said plurality of images, is a tensor.
8. The method according to claim 7 , wherein each of said plurality of tensors represent a color image.
9. The method according to claim 1 , wherein said numerical operator comprises a representation of a discrete Fourier transform.
10. The method according to claim 9 , wherein said discrete Fourier transform comprises a one-dimensional transformation acting on vectors.
11. The method according to claim 9 , wherein said discrete Fourier transform comprises a two-dimensional transformation acting on matrices.
12. The method according to claim 11 , wherein said two-dimensional discrete Fourier transform is represented as a tensor product of two one-dimensional discrete Fourier transforms, wherein a first one of said two one-dimensional discrete Fourier transforms is uniquely associated with rows of said data array, and a second one of said two one-dimensional discrete Fourier transforms is uniquely associated with columns of said data array.
13. The method according to claim 1 , wherein said numerical operator comprises a matrix.
14. The method according to claim 13 , wherein said ordered similarity transformation operator (P) comprises an array formed from eigenvectors of said numerical operator.
15. The method according to claim 14 , wherein said correspondingly-ordered diagonal operator (Λ) comprises a diagonal matrix of eigenvalues corresponding to said eigenvectors which are arranged in corresponding order relative to said eigenvalues.
16. The method according to claim 1 , wherein said multiplying accomplished in operations (b), (d), (e), and (f) is accomplished using matrix multiplication.
17. The method according to claim 1 , wherein said numerical operator comprises a tensor.
18. The method according to claim 1 , wherein said ordered similarity transformation operator (P) comprises a matrix.
19. The method according to claim 1 , wherein said multiplying of said approximate inverse of said ordered similarity transformation operator and said correspondingly-ordered diagonal operator is realized by multiplying each column of said ordered similarity transformation operator by a value of a diagonal element of a corresponding row of said correspondingly-ordered diagonal operator.
20. The method according to claim 1 , wherein said numerical operator comprises a zero origin which matches a zero origin of each of said plurality of images.
21. The method according to claim 20 , said method further comprising: shifting an index of said numerical operator so that said zero origin of said numerical operator matches said zero origin of each of said plurality of images.
22. The method according to claim 20 , wherein said plurality of images form a composite image.
23. A method for approximating the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α) having any one of a plurality of values, said method comprising: (a) representing an image using a data array; (b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P −1 ) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to one of said plurality of values of said power (α) to produce a fractional power diagonal operator; (d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P −1 ) to produce a first partial result; (e) multiplying said data array by said ordered similarity transformation operator (P) to produce a modified data array; (f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator; and (g) repeating operations (c) and (d) for each of said plurality of values of said power (α).
24. The method according to claim 23 , wherein said data array, for each of said plurality of images, is a vector.
25. The method according to claim 23 , wherein said data array, for each of said plurality of images, is a matrix.
26. The method according to claim 25 , wherein each of said plurality of matrices represent a monochrome image.
27. The method according to claim 25 , wherein each of said plurality of matrices represent a luminance component of a represented image.
28. The method according to claim 25 , wherein each of said plurality of matrices represent a chroma component of a represented image.
29. The method according to claim 23 , wherein said data array, for each of said plurality of images, is a tensor.
30. The method according to claim 29 , wherein each of said plurality of tensors represent a color image.
31. The method according to claim 23 , wherein said numerical operator comprises a representation of a discrete Fourier transform.
32. The method according to claim 31 , wherein said discrete Fourier transform comprises a one-dimensional transformation acting on vectors.
33. The method according to claim 31 , wherein said discrete Fourier transform comprises a two-dimensional transformation acting on matrices.
34. The method according to claim 33 , wherein said two-dimensional discrete Fourier transform is represented as a tensor product of two one-dimensional discrete Fourier transforms, wherein a first one of said two one-dimensional discrete Fourier transforms is uniquely associated with rows of said data array, and a second one of said two one-dimensional discrete Fourier transforms is uniquely associated with columns of said data array.
35. The method according to claim 23 , wherein said numerical operator comprises a matrix.
36. The method according to claim 35 , wherein said ordered similarity transformation operator (P) comprises an array formed from eigenvectors of said numerical operator.
37. The method according to claim 36 , wherein said correspondingly-ordered diagonal operator (Λ) comprises a diagonal matrix of eigenvalues corresponding to said eigenvectors which are arranged in corresponding order relative to said eigenvalues.
38. The method according to claim 23 , wherein said multiplying accomplished in operations (b), (d), (e), and (f) is accomplished using matrix multiplication.
39. The method according to claim 23 , wherein said numerical operator comprises a tensor.
40. The method according to claim 23 , wherein said ordered similarity transformation operator (P) comprises a matrix.
41. The method according to claim 23 , wherein said multiplying of said approximate inverse of said ordered similarity transformation operator and said correspondingly-ordered diagonal operator is realized by multiplying each column of said ordered similarity transformation operator by a value of a diagonal element of a corresponding row of said correspondingly-ordered diagonal operator.
42. The method according to claim 23 , wherein said numerical operator comprises a zero origin which matches a zero origin of said image.
43. The method according to claim 42 , said method further comprising: shifting an index of said numerical operator so that said zero origin of said numerical operator matches said zero origin of said image.
44. The method according to claim 42 , wherein said image is one portion of a composite image.
45. A computer-readable medium containing instructions for controlling a computer system to approximate the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α), said controlling provided by said computer system being accomplished according to operations comprising: (a) representing a plurality of images using an individual data array for each of said plurality of images; (b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P −1 ) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to said power (α) to produce a fractional power diagonal operator; (d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P −1 ) to produce a first partial result; (e) multiplying a data array of one of said plurality of images by said ordered similarity transformation operator (P) to produce a modified data array; (f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator; and (g) repeating operations (e) and (f) for each of said plurality of images.
46. A computer-readable medium containing instructions for controlling a computer system to approximate the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α) having any one of a plurality of values, said controlling provided by said computer system being accomplished according to operations comprising: (a) representing an image using a data array; (b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P −1 ) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to one of said plurality of values of said power (α) to produce a fractional power diagonal operator; (d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P −1 ) to produce a first partial result; (e) multiplying said data array by said ordered similarity transformation operator (P) to produce a modified data array; (f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator; and (g) repeating operations (c) and (d) for each of said plurality of values of said power (α).
Unknown
May 2, 2006
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